Masteries · Jason GlowneyArithmoí Book 1

Masteries

1.1.2 Matrix Group Properties

Jason Glowney

Clo­sure: The prod­uct of el­e­ments in the group is an­oth­er el­e­ment in the group

Closure:  ∀ gj,gk∈G, gj ∙gk∈G.\begin{array}{r} Closure:\ \ \forall\ g_{j},g_{k} \in G,\ g_{j}\ \bullet g_{k} \in G. \end{array}
(1.1.1)

As­so­cia­tiv­i­ty: The group­ing of mul­ti­pli­ca­tion of a linear com­bi­na­tion of el­e­ments yields the same result

Associativity:  ∀ gj,gk,gl∈G,( gj ∙gk)∙gl=gj∙(gk∙gl).\begin{array}{r} Associativity:\ \ \forall\ g_{j},g_{k,}g_{l} \in G,\left( \ g_{j}\ \bullet g_{k} \right) \bullet g_{l} = g_{j} \bullet \left( g_{k} \bullet g_{l} \right). \end{array}
(1.1.2)

Iden­ti­ty: There is an el­e­ment in the group when mul­ti­plied on the right, or the left, with any other el­e­ment in the group re­sults in the latter el­e­ment

Identity:  ∃  e∈G:∀ gj∈G, gj∙e=e∙gj=gj.\begin{array}{r} Identity:\ \ \exists\ \ e \in G:\forall\ g_{j} \in G,\ g_{j} \bullet e = e \bullet g_{j} = g_{j}. \end{array}
(1.1.3)

In­verse El­e­ment: For each el­e­ment in the group there exists an in­verse el­e­ment, for which, when both el­e­ments are taken to­geth­er as a prod­uct, the Iden­ti­ty el­e­ment is the result

Inverse Element: ∀ gj∈G ∃ gk∈G:gj∙gk=gk∙gj=e.\begin{array}{r} Inverse\ Element:\ \forall\ g_{j} \in G\ \exists\ g_{k} \in G:g_{j} \bullet g_{k} = g_{k} \bullet g_{j} = e. \end{array}
(1.1.4)

We will focus on finite groups, which rep­re­sent a closed set of in­vert­ible per­mu­ta­tions, while mind­ful of these de­f­i­n­i­tions. The per­mu­ta­tions of finite groups will be con­sid­ered using the square matrix form to better ap­pre­ci­ate di­men­sion­al­i­ty and other geo­met­ric un­der­pin­nings. Sev­er­al Matrix group de­f­i­n­i­tions exist, and we begin with one of the more en­com­pass­ing and straight­for­ward ex­am­ples, the gen­er­al linear group.